Why is this happening in my UV Unwrapping?
You do not have seams on the eye spheres, so either add seams, or try unwrapping with Smart UV Project or Sphere Projection.

What you are seeing is Blender unwrapping every individual face on top of each other.

I would highly recommend that you read the Blender manual section on unwrapping before continuing.

Also, see this question if you want details on unwrapping spheres in particular: 

What is the best way to unwrap a sphere?